Abstract
We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS 2 classical spins asL→∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has “diagonal form”).
Similar content being viewed by others
References
Adams, J.F.: Lectures on Lie groups. New York: Benjamin 1969
Asano, T.: Prog. Theo. Phys.40, 1328 (1968); J. Phys. Soc. Japan25, 1220 (1968)
Berezin, F.: Math. USSR 5688, 269 (1972)
Berezin, F.: Math. USSR Izv.6, 117 (1972)
Berezin, F.: Commun. Math. Phys.40, 153 (1975)
Boerner, H.: Representations of groups. Amsterdam: North Holland 1970
Combes, J., Schrader, R., Seiler, R.: Ann. Phys.111, 1 (1978)
Dunlop, F., Newman, C.: Commun. math. Phys.44, 223 (1975)
Fuller, W., Lenard, A.: Commun. math. Phys.69, 99 (1979)
Gilmore, R.: Ann. Phys.74, 391 (1972)
Gilmore, R.: Rev. Mex. de Fis.23, 143 (1974)
Gilmore, R.: J. Math. Phys.20, 891 (1979)
Golden, S.: Phys. Rev.137, 1127 (1965)
Griffiths, R.: J. Math. Phys.10, 1559 (1969)
Hepp, K., Lieb, E.: Phys. Rev.8A, 2517 (1973)
Hess, H., Schrader, R., Uhlenbrock, D.: Duke Math. J.44, 893 (1977)
Kato, T.: In: Topics in analysis, pp. 185–195. London, New York: Academic Press 1978
Klauder, J.: Ann. Phys.11, 123 (1960)
Klauder, J.: J. Math. Phys.5, 177 (1964)
Kostant, B.: Lecture notes in mathematics170, p. 87. Berlin, Heidelberg, New York: Springer 1970
Kostant, B.: Private communication
Levin, S.: Private communication
Lieb, E.: Commun. math. Phys.31, 327 (1973)
Lieb, E., Simon, B.: Adv. Math.23, 22–116 (1977)
Parthasarathy, K., Ranga Rao, R., Varadarajan, V.S.: Ann. Math.85, 383 (1967)
Perelomov, S.: Commun. math. Phys.26, 222 (1972)
Reed, M., Simon, B.: Methods of modern mathematical physics, IV. Analysis of operators. New York, London: Academic Press 1978
Robinson, D.:The thermodynamic pressure in quantum statistical mechanics. Berlin, Heidelberg, New York: Springer 1971
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
Samelson, H.: Notes on Lie algebras. Wokingham: van Nostrand 1969
Simon, B.: Ind. Math. J.26, 1067 (1977)
Simon, B.: J. Func. Anal.32, 97 (1979)
Simon, B.: J. Op. Theory,1, 37 (1979)
Simon, B.: Adv. Math.30, 268 (1978)
Simon, B.: Trace ideal and their applications. Cambridge: Cambridge University Press, 1979
Simon, B.: Functional integration and quantum physics. New York, London: Academic Press 1979
Simon, B.: In: Proc. 1979 Random Fields Colloq., Hungary
Souriau, J.: Structures des systèmes dynamiques. Paris: Dunod 1972
Suzuki, M., Fisher, M.: J. Math. Phys.12, 235 (1971)
Thirring, W.: Lectures on physics, Vol. 4; Univ. of Vienna Mimeographed notes; German publication expected 1980; English translation, 1981
Thompson, C.: J. Math. Phys.6, 1812 (1965); Ind. Math. J.21, 469 (1971)
Wehrl, A.: Rep. Math. Phys.12, 385 (1977)
Baumgartner, B.: Commun. Math. Phys. to appear
Author information
Authors and Affiliations
Additional information
Communicated by E. Lieb
Supported by USNSF Grant MCS-78-01885
Rights and permissions
About this article
Cite this article
Simon, B. The classical limit of quantum partition functions. Commun.Math. Phys. 71, 247–276 (1980). https://doi.org/10.1007/BF01197294
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01197294